Convolution and Equidistribution : : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / / Nicholas M. Katz.
Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new direction...
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Place / Publishing House: | Princeton, NJ : : Princeton University Press, , [2012] ©2012 |
Year of Publication: | 2012 |
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Language: | English |
Series: | Annals of Mathematics Studies ;
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Katz, Nicholas M., author. aut http://id.loc.gov/vocabulary/relators/aut Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / Nicholas M. Katz. Course Book Princeton, NJ : Princeton University Press, [2012] ©2012 1 online resource (208 p.) text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Annals of Mathematics Studies ; 180 Frontmatter -- Contents -- Introduction -- CHAPTER 1. Overview -- CHAPTER 2. Convolution of Perverse Sheaves -- CHAPTER 3. Fibre Functors -- CHAPTER 4. The Situation over a Finite Field -- CHAPTER 5. Frobenius Conjugacy Classes -- CHAPTER 6. Group-Theoretic Facts about Ggeom and Garith -- CHAPTER 7. The Main Theorem -- CHAPTER 8. Isogenies, Connectedness, and Lie-Irreducibility -- CHAPTER 9. Autodualities and Signs -- CHAPTER 10. A First Construction of Autodual Objects -- CHAPTER 11. A Second Construction of Autodual Objects -- CHAPTER 12. The Previous Construction in the Nonsplit Case -- CHAPTER 13. Results of Goursat-Kolchin-Ribet Type -- CHAPTER 14. The Case of SL(2); the Examples of Evans and Rudnick -- CHAPTER 15. Further SL(2) Examples, Based on the Legendre Family -- CHAPTER 16. Frobenius Tori and Weights; Getting Elements of Garith -- CHAPTER 17. GL(n) Examples -- CHAPTER 18. Symplectic Examples -- CHAPTER 19. Orthogonal Examples, Especially SO(n) Examples -- CHAPTER 20. GL(n) x GL(n) x ... x GL(n) Examples -- CHAPTER 21. SL(n) Examples, for n an Odd Prime -- CHAPTER 22. SL(n) Examples with Slightly Composite n -- CHAPTER 23. Other SL(n) Examples -- CHAPTER 24. An O(2n) Example -- CHAPTER 25. G2 Examples: the Overall Strategy -- CHAPTER 26. G2 Examples: Construction in Characteristic Two -- CHAPTER 27. G2 Examples: Construction in Odd Characteristic -- CHAPTER 28. The Situation over ℤ: Results -- CHAPTER 29. The Situation over ℤ: Questions -- CHAPTER 30. Appendix: Deligne's Fibre Functor -- Bibliography -- Index restricted access http://purl.org/coar/access_right/c_16ec online access with authorization star Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject. Mode of access: Internet via World Wide Web. In English. Description based on online resource; title from PDF title page (publisher's Web site, viewed 31. Jan 2022) Convolutions (Mathematics). Mellin transform. Sequences (Mathematics). MATHEMATICS / Number Theory. bisacsh ArtinГchreier reduced polynomial. Emanuel Kowalski. EulerАoincar formula. Frobenius conjugacy class. Frobenius conjugacy. Frobenius tori. GoursatЋolchinВibet theorem. Kloosterman sheaf. Laurent polynomial. Legendre. Pierre Deligne. Ron Evans. Tannakian category. Tannakian groups. Zeeev Rudnick. algebro-geometric. autodual objects. autoduality. characteristic two. connectedness. dimensional objects. duality. equidistribution. exponential sums. fiber functor. finite field Mellin transform. finite field. finite fields. geometrical irreducibility. group scheme. hypergeometric sheaf. interger monic polynomials. isogenies. lie-irreducibility. lisse. middle convolution. middle extension sheaf. monic polynomial. monodromy groups. noetherian connected scheme. nonsplit form. nontrivial additive character. number theory. odd characteristic. odd prime. orthogonal case. perverse sheaves. polynomials. pure weight. semisimple object. semisimple. sheaves. signs. split form. supermorse. theorem. theorems. Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 9783110494914 ZDB-23-PMB Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013 9783110442502 https://doi.org/10.1515/9781400842704?locatt=mode:legacy https://www.degruyter.com/isbn/9781400842704 Cover https://www.degruyter.com/document/cover/isbn/9781400842704/original |
language |
English |
format |
eBook |
author |
Katz, Nicholas M., Katz, Nicholas M., |
spellingShingle |
Katz, Nicholas M., Katz, Nicholas M., Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / Annals of Mathematics Studies ; Frontmatter -- Contents -- Introduction -- CHAPTER 1. Overview -- CHAPTER 2. Convolution of Perverse Sheaves -- CHAPTER 3. Fibre Functors -- CHAPTER 4. The Situation over a Finite Field -- CHAPTER 5. Frobenius Conjugacy Classes -- CHAPTER 6. Group-Theoretic Facts about Ggeom and Garith -- CHAPTER 7. The Main Theorem -- CHAPTER 8. Isogenies, Connectedness, and Lie-Irreducibility -- CHAPTER 9. Autodualities and Signs -- CHAPTER 10. A First Construction of Autodual Objects -- CHAPTER 11. A Second Construction of Autodual Objects -- CHAPTER 12. The Previous Construction in the Nonsplit Case -- CHAPTER 13. Results of Goursat-Kolchin-Ribet Type -- CHAPTER 14. The Case of SL(2); the Examples of Evans and Rudnick -- CHAPTER 15. Further SL(2) Examples, Based on the Legendre Family -- CHAPTER 16. Frobenius Tori and Weights; Getting Elements of Garith -- CHAPTER 17. GL(n) Examples -- CHAPTER 18. Symplectic Examples -- CHAPTER 19. Orthogonal Examples, Especially SO(n) Examples -- CHAPTER 20. GL(n) x GL(n) x ... x GL(n) Examples -- CHAPTER 21. SL(n) Examples, for n an Odd Prime -- CHAPTER 22. SL(n) Examples with Slightly Composite n -- CHAPTER 23. Other SL(n) Examples -- CHAPTER 24. An O(2n) Example -- CHAPTER 25. G2 Examples: the Overall Strategy -- CHAPTER 26. G2 Examples: Construction in Characteristic Two -- CHAPTER 27. G2 Examples: Construction in Odd Characteristic -- CHAPTER 28. The Situation over ℤ: Results -- CHAPTER 29. The Situation over ℤ: Questions -- CHAPTER 30. Appendix: Deligne's Fibre Functor -- Bibliography -- Index |
author_facet |
Katz, Nicholas M., Katz, Nicholas M., |
author_variant |
n m k nm nmk n m k nm nmk |
author_role |
VerfasserIn VerfasserIn |
author_sort |
Katz, Nicholas M., |
title |
Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / |
title_sub |
Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / |
title_full |
Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / Nicholas M. Katz. |
title_fullStr |
Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / Nicholas M. Katz. |
title_full_unstemmed |
Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / Nicholas M. Katz. |
title_auth |
Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / |
title_alt |
Frontmatter -- Contents -- Introduction -- CHAPTER 1. Overview -- CHAPTER 2. Convolution of Perverse Sheaves -- CHAPTER 3. Fibre Functors -- CHAPTER 4. The Situation over a Finite Field -- CHAPTER 5. Frobenius Conjugacy Classes -- CHAPTER 6. Group-Theoretic Facts about Ggeom and Garith -- CHAPTER 7. The Main Theorem -- CHAPTER 8. Isogenies, Connectedness, and Lie-Irreducibility -- CHAPTER 9. Autodualities and Signs -- CHAPTER 10. A First Construction of Autodual Objects -- CHAPTER 11. A Second Construction of Autodual Objects -- CHAPTER 12. The Previous Construction in the Nonsplit Case -- CHAPTER 13. Results of Goursat-Kolchin-Ribet Type -- CHAPTER 14. The Case of SL(2); the Examples of Evans and Rudnick -- CHAPTER 15. Further SL(2) Examples, Based on the Legendre Family -- CHAPTER 16. Frobenius Tori and Weights; Getting Elements of Garith -- CHAPTER 17. GL(n) Examples -- CHAPTER 18. Symplectic Examples -- CHAPTER 19. Orthogonal Examples, Especially SO(n) Examples -- CHAPTER 20. GL(n) x GL(n) x ... x GL(n) Examples -- CHAPTER 21. SL(n) Examples, for n an Odd Prime -- CHAPTER 22. SL(n) Examples with Slightly Composite n -- CHAPTER 23. Other SL(n) Examples -- CHAPTER 24. An O(2n) Example -- CHAPTER 25. G2 Examples: the Overall Strategy -- CHAPTER 26. G2 Examples: Construction in Characteristic Two -- CHAPTER 27. G2 Examples: Construction in Odd Characteristic -- CHAPTER 28. The Situation over ℤ: Results -- CHAPTER 29. The Situation over ℤ: Questions -- CHAPTER 30. Appendix: Deligne's Fibre Functor -- Bibliography -- Index |
title_new |
Convolution and Equidistribution : |
title_sort |
convolution and equidistribution : sato-tate theorems for finite-field mellin transforms (am-180) / |
series |
Annals of Mathematics Studies ; |
series2 |
Annals of Mathematics Studies ; |
publisher |
Princeton University Press, |
publishDate |
2012 |
physical |
1 online resource (208 p.) |
edition |
Course Book |
contents |
Frontmatter -- Contents -- Introduction -- CHAPTER 1. Overview -- CHAPTER 2. Convolution of Perverse Sheaves -- CHAPTER 3. Fibre Functors -- CHAPTER 4. The Situation over a Finite Field -- CHAPTER 5. Frobenius Conjugacy Classes -- CHAPTER 6. Group-Theoretic Facts about Ggeom and Garith -- CHAPTER 7. The Main Theorem -- CHAPTER 8. Isogenies, Connectedness, and Lie-Irreducibility -- CHAPTER 9. Autodualities and Signs -- CHAPTER 10. A First Construction of Autodual Objects -- CHAPTER 11. A Second Construction of Autodual Objects -- CHAPTER 12. The Previous Construction in the Nonsplit Case -- CHAPTER 13. Results of Goursat-Kolchin-Ribet Type -- CHAPTER 14. The Case of SL(2); the Examples of Evans and Rudnick -- CHAPTER 15. Further SL(2) Examples, Based on the Legendre Family -- CHAPTER 16. Frobenius Tori and Weights; Getting Elements of Garith -- CHAPTER 17. GL(n) Examples -- CHAPTER 18. Symplectic Examples -- CHAPTER 19. Orthogonal Examples, Especially SO(n) Examples -- CHAPTER 20. GL(n) x GL(n) x ... x GL(n) Examples -- CHAPTER 21. SL(n) Examples, for n an Odd Prime -- CHAPTER 22. SL(n) Examples with Slightly Composite n -- CHAPTER 23. Other SL(n) Examples -- CHAPTER 24. An O(2n) Example -- CHAPTER 25. G2 Examples: the Overall Strategy -- CHAPTER 26. G2 Examples: Construction in Characteristic Two -- CHAPTER 27. G2 Examples: Construction in Odd Characteristic -- CHAPTER 28. The Situation over ℤ: Results -- CHAPTER 29. The Situation over ℤ: Questions -- CHAPTER 30. Appendix: Deligne's Fibre Functor -- Bibliography -- Index |
isbn |
9781400842704 9783110494914 9783110442502 |
callnumber-first |
Q - Science |
callnumber-subject |
QA - Mathematics |
callnumber-label |
QA432 |
callnumber-sort |
QA 3432 K38 42017 |
url |
https://doi.org/10.1515/9781400842704?locatt=mode:legacy https://www.degruyter.com/isbn/9781400842704 https://www.degruyter.com/document/cover/isbn/9781400842704/original |
illustrated |
Not Illustrated |
dewey-hundreds |
500 - Science |
dewey-tens |
510 - Mathematics |
dewey-ones |
515 - Analysis |
dewey-full |
515.723 |
dewey-sort |
3515.723 |
dewey-raw |
515.723 |
dewey-search |
515.723 |
doi_str_mv |
10.1515/9781400842704?locatt=mode:legacy |
oclc_num |
979905293 |
work_keys_str_mv |
AT katznicholasm convolutionandequidistributionsatotatetheoremsforfinitefieldmellintransformsam180 |
status_str |
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ids_txt_mv |
(DE-B1597)447783 (OCoLC)979905293 |
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hierarchy_parent_title |
Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Backlist 2000-2013 |
is_hierarchy_title |
Convolution and Equidistribution : Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180) / |
container_title |
Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020 |
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tag="653" ind1=" " ind2=" "><subfield code="a">supermorse.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">theorem.</subfield></datafield><datafield tag="653" ind1=" " ind2=" "><subfield code="a">theorems.</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Title is part of eBook package:</subfield><subfield code="d">De Gruyter</subfield><subfield code="t">Princeton Annals of Mathematics eBook-Package 1940-2020</subfield><subfield code="z">9783110494914</subfield><subfield code="o">ZDB-23-PMB</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Title is part of eBook package:</subfield><subfield code="d">De Gruyter</subfield><subfield code="t">Princeton University Press eBook-Package Backlist 2000-2013</subfield><subfield code="z">9783110442502</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield 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